By Cornelius Lanczos
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Additional resources for Linear differential operators.
1 Example Suppose the weekly levels of two products to be manufactured are to be determined. Suppose our first criterion relates to the amount of hours labour per week used to manufacture the two products A and B. Assume our decision variables are x1 : The number of products of type A manufactured per week x1 : The number of products of type B manufactured per week and each type A product takes 4 h to manufacture and each type B product takes 3 h to manufacture. Our function for the amount of labour used is given by f1 (x) = 4x1 + 3x2 Now in order to complete the goal formulation the two deviational variables must be added and a goal level set.
Refer to Applegate et al. (2007) for details. Example 7 – Downstream Oil Industry An oil company has to transport oil products from three sites (S1, S2, S3), through three refineries (R1, R2, R3), onto three depots (D1, D2, D3), and finally to three customers (C1, C2, C3). 17. 18. 19. 20. 20 Depot to customer costs • • • • From / to C1 C2 C3 D1 D2 D3 6 4 5 8 10 9 5 7 8 Goal 1: Keep costs to below £40,000 per day. Goal 2: Keep each refinery operating at at least 75% of its capacity. Goal 3: Keep each depot at a level of between 50 and 90% of its capacity.
For example if the company has a strategic aim to produce 40 of each of the two products but only machine time for 75 products then the strategic aim will never be met. Other conflicts are less obvious. For example Fig. 1 shows that no solution exists that simultaneously meets the target levels for labour and for profit. A level of conflict between goals is to be expected in any modern organisation due to the laudable aim to achieve good levels on all criteria. 2 Variant Choice After the formulation of the goals, constraints, and sign restrictions the achievement function must now be formulated.
Linear differential operators. by Cornelius Lanczos